Guided matter-wave Sagnac interferometer

ABSTRACT

The present invention provides an interferometer apparatus comprising a matter-wave guide enclosing an area, wherein a flux of particles may be guided in the matter-wave guide in at least two opposite paths, the matter-wave guide is rotatable relative to an inertial frame of reference; a first beam splitter to split the first beam to at least second and third beams, each of the second and third beams is to be guided in another path of the two opposite paths; and a second beam splitter allowing particles of the second and third beams to exit the matter-wave guide in a first probability and to stay in the matter-wave guide in a second probability.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a National Phase Application of PCT InternationalApplication No. PCT/IL2007/001611, entitled “GUIDED MATTER-WAVE SAGNACINTERFEROMETER”, International Filing Date Dec. 26, 2007, published onJul. 10, 2008 as International Publication No. WO 2008/081431, which inturn claims the benefit of U.S. Provisional Patent Application Ser. No.60/877,591, filed Dec. 29, 2006, both of which are incorporated in theirentirety herein by reference.

BACKGROUND OF THE INVENTION

Sagnac effect is a phase shift induced in a wave propagating in a loopby rotation of the loop. Two waves propagating in opposite directionsalong a rotating closed loop may interfere in a point of exit, whereinthe rotation may cause a phase shift between them. An optical SagnacInterferometer apparatus may include, for example, a loop trajectoryenclosing an area. The interferometer may operate by splitting a beam oflight at an entrance of the loop, to two beams which may be made topropagate in opposite directions along the loop trajectory. Aninterference pattern may be obtained at a point of exit from the loop.When the apparatus rotates, the path length of one of the beams may beeffectively shortened, while the path length of the other beam may beeffectively lengthened. Therefore, a phase shift may be created betweenthe two beams, dependent on the rotation velocity of the apparatus.

Therefore, a Sagnac interferometer may measure changes in the rotationfrequency of the frame that it is fixed to, relative to a global(inertial) frame of reference. The Sagnac effect in a wave propagatingthrough a closed rotating ring induces a phase shift proportional to therotation frequency Ω of this rotation and the area A of the ring. Forlight waves with frequency ω this phase shift may be represented byφ_(light)=(2ωA/c²)Ω, where c is the speed of light and w is the lightfrequency. For matter waves of massive particles with mass m, theinduced phase shift may be represented by φ_(matter)=(2 mA/ℏ)Ω, which islarger than the phase shift in an optical SI having the same area A bymc²/ℏω, wherein ℏ is planck's constant. Therefore, the Sagnac phase ismuch more sensitive to rotations in a matter-wave SI compared to lightwave SI, for example, by 10 orders of magnitude. However,state-of-the-art matter-wave Sagnac interferometers, which are based onbeams of atoms traveling in free space, are limited by their area numberof atoms and the allowed momentum bandwidth of the atoms.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter regarded as the invention is particularly pointed outand distinctly claimed in the concluding portion of the specification.The invention, however, both as to organization and method of operation,together with objects, features, and advantages thereof, may best beunderstood by reference to the following detailed description when readwith the accompanying drawings in which:

FIG. 1 is a schematic illustration of a SI (Sagnac interferometer)according to some embodiments of the present invention;

FIG. 2 is an exemplary illustration of a SI (Sagnac interferometer)according to some embodiments of the present invention; and

FIG. 3 is a graph illustrating output transmission probability of a SI(Sagnac interferometer) according to some embodiments of the presentinvention.

It will be appreciated that for simplicity and clarity of illustration,elements shown in the figures have not necessarily been drawn to scale.For example, the dimensions of some of the elements may be exaggeratedrelative to other elements for clarity. Further, where consideredappropriate, reference numerals may be repeated among the figures toindicate corresponding or analogous elements.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

In the following detailed description, numerous specific details are setforth in order to provide a thorough understanding of the invention.However, it will be understood by those skilled in the art that thepresent invention may be practiced without these specific details. Inother instances, well-known methods, procedures, and components have notbeen described in detail so as not to obscure the present invention.

The present invention may provide a Sagnac interferometer employingguided matter wave. The interferometer according to embodiments of thepresent invention may operate with high flux of particles with widebandwidth, thus providing a clear and evident signal over transientnoise. The interferometer according to embodiments of the presentinvention may provide better rotation sensitivity than priorinterferometers by using matter-waves instead of light waves. Theinterferometer according to embodiments of the present invention mayfurther provide better rotation sensitivity relative to for example,previous matter-wave SIs (Sagnac interferometers) by providing largereffective area of the SI loop, for example, without enlargement of thereal area occupied by the loop.

Reference is now made to FIG. 1, which is a schematic illustration of aSI (Sagnac interferometer) 100 according to some embodiments of thepresent invention. SI 100 may include a loop guide 110, along whichmatter-waves of particles may propagate in clockwise or counterclockwisedirection, for example, in trajectories a or b respectively. SI 100 mayinclude input port i, through which, for example, matter-waves may enterthe loop. Input i may be connected to loop 110 by splitter 130, whichmay split the matter-wave, for example, upon entrance, to twomatter-waves propagating in opposite directions, for example, throughloop ports α and β. SI 100 may include output port o, through which theparticles may exit loop 110. Output port o may be connected to loop 110by splitter 120, through which the particles may exit to output port oin a certain probability (1−R) or may stay in the loop trajectory in acomplimentary probability R. As the probability R is larger, theparticles may do more passes through loop 110 before leaving loop 110through output port o. The paths of the particles through port α or β tosplitter 120 may have lengths L_(α) or L_(β), respectively. IfL_(α)≠L_(β), there may be a phase difference between the matter-waves intrajectories a and b, which may be proportional to the velocity of theparticles in the matter wave. Loop 110 may rotate at a rotationfrequency Ω relative to inertial frame 140. If loop 110 rotates, forexample, clockwise, particles propagating clockwise in the loop throughtrajectory a may reach the splitter 120 with an additional negativephase shift, while particles propagating anti clockwise throughtrajectory b may reach splitter 120 with a positive phase shift. Thephase difference between the two trajectories may determine the outputamplitude a_(out). Thus, the phase shift between the matter-waves intrajectories a and b may change with the rotation frequency Ω. Therotation velocity Ω may be deduced, therefore, from the output amplitudea_(out). In a general embodiment of an SI, if other origins for velocity(or momentum) dependent phase shifts between the two trajectories exist,such as, for example, when L_(α)≠L_(β), the interference pattern at theoutput port o may be smeared out and the rotation sensitivity maydisappear if the initial beam of particles is a mixture of manyvelocities.

SI 100 may receive input flux of particles through input port i, forexample, with momentum bandwidth Δk. If L_(α)≠L_(β) and Δk is relativelywide, e.g., ΔkδL>>1 (δL=L_(α)−L_(β)), the interferingcounter-propagating fluxes of wide range of longitudinal momentums fromports α and β may cancel each other at port o, so that, for example, thedependency of the output at port o on the rotational phase shiftφ_(matter) and/or the rotation frequency Ω may be substantiallyweakened. In this case, the rotational sensitivity of SI 100 may bedamaged. Only counter-propagating waves which follow trajectories withsubstantially the same length may contribute visible dependency in therotational phase shift in the output. In some embodiments of the presentinvention, in the case of L_(α)≠L_(β), internal reflections may enableexistence of opposite paths with same length.

In case most of the particles follow trajectories with substantially thesame length, e.g., the path lengths L_(α) and L_(β) are equal, Δk may berelatively wide without damaging the rotational sensitivity of SI 100.Thus, for example, high flux of particles with wide bandwidth isenabled, thus providing a clear signal which may be evident overtransient noise. This may facilitate very sensitive detection of smallchanges in rotation frequency Ω.

The area A is the effective area enclosed by the trajectory of theparticles. As mentioned above, particles may exit loop 110 throughsplitter 120 in a certain probability (1−R) or may stay in the looptrajectory in a complimentary probability R. As the probability R islarger, the particles may do more passes through loop 110 before leavingloop 110 through output port o. As the particles do more passes, theeffective area A is larger, thus, for example, the rotationalsensitivity of SI 100 may be greater.

Reference is now made to FIG. 2, which is an exemplary illustration of aSI (Sagnac interferometer) 200 according to some embodiments of thepresent invention. SI 200 may include a guiding loop 210, BS (beamsplitter) 220, BS 230, input port i and output port o. BS 230 may havesubstantially equal transmission and reflection probabilities, e.g.,which may be substantially equal to 0.5. BS 230 may split particle beamincident from input port i into two beams, e.g., reflected beam andtransmitted beam, with substantially equal probabilities and, forexample, a phase difference of approximately π/2 between them. BS 220may have reflection probability r² which may be much higher than thetransmission probability t², such that r²+t²=1. BS 220 may split aparticle beam into two beams, e.g., reflected beam and transmitted beam,wherein, for example, the reflected beam may have much higherprobability than the transmitted beam and, for example, the phasedifference between them may be of approximately π/2. The transmissionand reflection amplitudes may be controllable by, for example, amagnetic tunneling bather and/or any other suitable means fortransmission of particle waves between channels with controllableprobability, for example, by using magnetic fields. As the reflectionprobability r² is higher than the transmission probability t², aparticle may propagate in loop 210 for a large number of times, thus,for example, enlarging the effective area A and/or the rotationalsensitivity of SI 100.

Loop 210 may rotate at a rotation frequency Ω relative to inertial frame240. As described above with reference to FIG. 1, if loop 210 rotates,for example, clockwise, particles propagating clockwise in the loop mayreach the splitter 220 with an additional negative phase shift, whileparticles propagating anti clockwise through trajectory b may reachsplitter 220 with a positive phase shift. The phase difference betweenthe two trajectories may determine the output amplitude a_(out). Thus,for example, the phase shift between the matter-waves in trajectories αand β may change with the rotation frequency Ω. The rotation frequency Ωmay be deduced, therefore, from the output amplitude a_(out).

According to some embodiments of the present invention, a particle, forexample, an atom, incident from port i may exit at port o through one offour kinds of trajectories: (a) transmission through BS 230, reflectionat BS 220 and then transmission through BS 230 again. The total excessphase which may be gained by this path is approximately π; (b)reflection at BS 230, reflection at BS 220 and then reflection at BS 230again, with substantially no excess phase; (c) transmission at BS 230,transmission at BS 220, propagation in the counterclockwise directionthrough loop 210 for a certain number of times and then transmissionagain through BS 220 and then through BS 230, with a total excess phaseof approximately 2π and a phase due to Sagnac effect because of thepropagation through loop 210; and (d) reflection at BS 230, transmissionat BS 220, propagation in clockwise direction in loop 210 for a certainnumber of times and then transmission again through BS 220 and thenthrough BS 230, with total phase of approximately n and a phase due toSagnac effect because of the propagation through loop 210. Trajectories(a) and (b) and trajectories (c) and (d) may have substantially the samelength.

When loop 210 is not rotating relative to inertial frame 240 the phasein trajectories (a) and (b) may be substantially opposite and the phaseof trajectories (e) and (d) may be substantially opposite, thus, forexample, the probability to exit at port o may be substantially zero,due to, for example, full destructive interference. However, when loop210 is rotating relative to inertial frame 240 the particle beamspropagating clockwise and counterclockwise through loop 210 may addopposite phase shifts proportional to the rotation frequency Ω and tothe number of passes through loop 210. These phase shifts may increasethe probability for a positive output of particles at port o, whereinthe probability may depend on the rotation frequency Ω. Therefore, forexample, the rotation frequency Ω and/or changes in the rotationfrequency Ω may be deduced by measurement of output transmission at porto.

In the embodiment of the present invention shown in FIG. 2, a largemajority of the particles may follow trajectories with substantially thesame length, for example, trajectories (c) and (d) described in detailabove. Therefore, it may be possible to use SI 200 with high flux ofparticles with wide momentum bandwidth, substantially without decreasingthe rotational sensitivity of SI 200.

In this embodiments of the present invention, for example, the outputtransmission probability at port o as a function of the rotational phaseφ(Ω), which may be notated P(φ), may be approximately represented by

${\overset{\_}{P}(\phi)} = {{\frac{t^{2}}{1 + r^{2}}\left\lbrack {1 - \frac{\cos^{2}\phi}{1 + {4\left( {r/t^{2}} \right)^{2}\sin^{2}\phi}}} \right\rbrack}.}$

Accordingly, for example, the output transmission probability P(φ) maybe substantially zero when the rotational phase (Sagnac phase) φ issubstantially zero, e.g., when the rotation frequency Ω is substantiallyzero. When φ>>t²/r, the output transmission probability P(φ) mayasymptotically approach t²/2r. The thick curve in FIG. 3 is a graphillustrating the output transmission probability P(φ). Other curves inFIG. 3 illustrate output transmission probabilities of SIs according toembodiments of the present invention having imperfections in the guidingpotential of the loop, for example, with reflection amplitudes up to0.25. Thus, the dependency of output transmission probability P(φ) inrotation frequency Ω may be stronger as t/r is smaller.

The sensitivity of the output transmission at port o to changes in therotation frequency Ω in this embodiment of the present invention, may bedefined as the minimum change in the rotation frequency Ω which maygenerate a noticeable (beyond noise level) change in the outputtransmission at port o. This minimum change may be represented by

${{\delta\Omega}_{\min} = {\frac{h}{4\pi\;{{mAP}^{\prime}(\phi)}}\sqrt{\frac{P(\phi)}{N}}}},$wherein P′(φ) is the derivative of the output transmission probabilitywith respect to the Sagnac phase φ and N is the total number of incidentparticles integrated over the time of measurement. Thus, for example,the rotation sensitivity may be maximal when the Sagnac phase φ issubstantially zero, e.g., when the rotation frequency Ω is substantiallyzero. Additionally, for example, the rotation sensitivity may beproportional to the square root of the transmission amplitude t throughBS 220.

A Sagnac interferometer according to embodiments of the presentinvention may provide high rotational sensitivity due to, for example,the ability to propagate particles in loop 210 for a large number oftimes, thus, for example, enlarging the effective area A and/or therotational sensitivity of the Sagnac interferometer. Therefore, forexample, the actual size of a Sagnac interferometer according toembodiments of the present invention may be very small, for example,having a loop radius of 1 cm. This may enable, for example,implementation of such SI on an atom chip.

The maximum velocity of the atoms in this embodiment may be limited bythe magnetic force by which the SI may be bounded to an atom chip. Forexample, the SI may be bounded to the chip by magnetic field gradientsof approximate order of Gauss/μm, which may be generated, for example,by wires on the chip, for example, about 10 μm from the surface of thechip. The centrifugal force mv²/r of the circulating atoms should bebelow the magnetic force bounding the SI to the chip. For example, themaximum velocity of the atoms may be about 10 m/sec.

As described above with reference to FIG. 2, the time for which theatoms may stay in the SI may depend on the transmission probability t²of BS 220. The transmission amplitude of BS 220 may be, for example,about t=0.035. This transmission amplitude may allow an atom to stay inthe SI for about, e.g., 10 seconds, which may enable about 1000-2000rotations in the guiding loop. For an input flux of, for example, 10⁹atoms per second, a rotation sensitivity of about 5×10⁻¹²rad·s⁻¹/√{square root over (Hz)} may be obtained.

While certain features of the invention have been illustrated anddescribed herein, many modifications, substitutions, changes, andequivalents will now occur to those of ordinary skill in the art. It is,therefore, to be understood that the appended claims are intended tocover all such modifications and changes as fall within the true spiritof the invention.

1. An apparatus comprising: a matter-wave guide enclosing an area,wherein a flux of particles may be guided in said matter-wave guide inat least two opposite paths, said apparatus has a rotational velocityrelative to an inertial frame of reference; a first beam splitter tosplit a first beam to at least second and third beams, each of saidsecond and third beams is to be guided in another path of said twoopposite paths; and a second beam splitter allowing particles of saidsecond and third beams to exit said matter-wave guide in a firstprobability and to stay in said matter-wave guide in a secondprobability; wherein: said second beam splitter allows transmission ofparticles of said second and third beams in said first probability andreflection of particles of said second and third beams in said secondprobability; and said first probability is smaller than said secondprobability.
 2. The apparatus according to claim 1, wherein said firstbeam splitter allows transmission of particles of said first beam in athird probability and reflection of particles of said first beam in afourth probability.
 3. The apparatus according to claim 2, wherein saidthird and fourth probabilities are equal.
 4. The apparatus according toclaim 1, wherein said two opposite paths are equal in length.
 5. Theapparatus according to claim 1, wherein said second beam splitter allowsexit of particles of said second and third beams through an output portof said apparatus.
 6. The apparatus according to claim 5, wherein outputtransmission probability at said output port is substantially zero whenrotation frequency of said matter-wave guide is substantially zero. 7.The apparatus according to claim 5, wherein dependency of outputtransmission probability at said output port in rotation frequency ofsaid matter-wave guide is stronger as the ratio between said firstprobability and said second probability is smaller.
 8. An apparatuscomprising: a matter-wave guide enclosing an area, wherein a flux ofparticles may be guided in said matter-wave guide in at least twoopposite paths, said apparatus has a rotational velocity relative to aninertial frame of reference; a first beam splitter to split a first beamto at least second and third beams, each of said second and third beamsis to be guided in another path of said two opposite paths; and a secondbeam splitter allowing particles of said second and third beams to exitsaid matter-wave guide in a first probability and to stay in saidmatter-wave guide in a second probability; wherein: said second beamsplitter allows exit of particles of said second and third beams throughan output port of said apparatus; and a dependency of outputtransmission probability at said output port in rotation frequency ofsaid matter-wave guide is stronger as the ratio between said firstprobability and said second probability is smaller.
 9. The apparatusaccording to claim 8, wherein said first beam splitter allowstransmission of particles of said first beam in a third probability andreflection of particles of said first beam in a fourth probability. 10.The apparatus according to claim 9, wherein said third and fourthprobabilities are equal.
 11. The apparatus according to claim 8, whereinsaid second beam splitter allows transmission of particles of saidsecond and third beams in said first probability and reflection ofparticles of said second and third beams in said second probability. 12.The apparatus according to claim 11, wherein said first probability issmaller than said second probability.
 13. The apparatus according toclaim 8, wherein said two opposite paths are equal in length.
 14. Theapparatus according to claim 8, wherein an output transmissionprobability at said output port is substantially zero when a rotationfrequency of said matter-wave guide is substantially zero.